5

Accoustic Wave

P. K. Karmakar Department of Physics, Tezpur University, Napaam, Tezpur, Assam

India

1. Introduction

An acoustic wave basically is a mechanical oscillation of pressure that travels through a medium like solid, liquid, gas, or plasma in a periodic wave pattern transmitting energy from one point to another in the medium [1-2]. It transmits sound by vibrating organs in the ear that produce the sensation of hearing and hence, it is also called acoustic signal. This is well-known that air is a fluid. Mechanical waves in air can only be longitudinal in nature; and therefore, all sound waves traveling through air must be longitudinal waves originating in the transmission form of compression and rarefaction from vibrating matter in the medium. The propagation of sound in absence of any material medium is always impossible. Therefore, sound does not travel through the vacuum of outer space, since there is nothing to carry the vibrations from a source to a receiver. The nature of the molecules making up a substance determines how well or how rapidly the substance will carry sound waves. The two characteristic variables affecting the propagation of acoustic waves are (1) the inertia of the constituent molecules and (2) the strength of molecular interaction. Thus, hydrogen gas, with the least massive molecules, will carry a sound wave at 1,284.00 ms-1 when the gas temperature is 00 C [1]. More massive helium gas molecules have more inertia and carry a sound wave at only 965.00 ms-1 at the same temperature. A solid, however, has molecules that are strongly attached, so acoustic vibrations are passed rapidly from molecule to molecule. Steel, for an instant example, is highly elastic, and sound will move rapidly through a steel rail at 5,940.00 ms-1 at the same temperature. The temperature of a medium influences the phase speed of sound through it. The gas molecules in warmer air thus have a greater kinetic energy than those of cooler air. The molecules of warmer air therefore transmit an acoustic impulse from molecule to molecule more rapidly. More precisely, the speed of a sound wave increases by 0.60 ms-1 for each Celcius degree rise in temperature above 00 C. Acoustic waves, or sound waves, are defined generally and specified mainly by three characteristics: wavelength, frequency, and amplitude. The wavelength is the distance from the top of one wave’s crest to the next (or, from the top of one trough to the next). The frequency of a sound wave is the number of waves that pass a point each second [1]. Sound waves with higher frequencies have higher pitches than sound waves with lower frequencies and vice versa. Amplitude is the measure of energy in a sound wave and affects volume. The greater the amplitude of an acoustic wave, the louder the sound and vice versa. An acoustic wave is what makes humans and other animals able to hear. A person’s ear perceives the vibrations of an acoustic wave and interprets it as sound [1]. The outer ear, the visible part, is shaped like a funnel that collects sound waves and sends them into the ear

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canal where they hit the ear drum, which is a tightly stretched piece of skin that vibrates in time with the wave. The ear drum starts a chain reaction and sends the vibration through three little bones in the middle ear that amplify sound. Those bones are called the hammer, the anvil, and the stirrup. Furthermore, acoustic waves from a purely hydrodynamic point of view are small-amplitude disturbances that propagate in a compressible medium (like a fluid) through the interplay between fluid inertia, and the restoring force of fluid pressure. The propagation of small-amplitude disturbances in homogeneous medium is observed as acoustic waves such as water waves, and in self-gravitationally stratified medium like stellar atmosphere [36-37, 41-44], acoustic-gravity waves such as p-modes, g-modes, f-modes, etc., as found by helio- and astero-seismological studies. Acoustic waves propagating through a dispersive medium may get dynamically converted into solitons or shocks depending on the physical mechanisms responsible for their saturation. When fluid nonlinearity (convective effect) is balanced by dispersion (geometrical effect), solitons usually result [4]. Conversely, shocks are formed if fluid nonlinearity is balanced by dissipation (damping effect). The nonlinear hydrodynamic equations of various forms (like KdV equation, Burger equation, NLS equation, BO equation, etc.) in the context of the generation, structure, propagation, self-organization and dissipation of solitons or shocks have long been developed applying the hydrodynamic views of the usual conservation laws of flux, momentum and energy [4]. Similar outlook is needed to understand the formation of other nonlinear localized structures of low frequency acoustic waves like double layers, vortices, etc. They are important in a wide variety of space, astrophysical and laboratory problems for the investigation of dynamical stability against perturbation [3-4]. In addition, these equations have wide applications to study a nonlinear, radial, energetic, and steady-flow problem that provides a first rough approximation to the physics of stellar winds and associated acoustic wave kinetics, which are responsible for stellar mass-loss phenomena via supersonic flow into interstellar space [2]. Acoustic mode in plasmas of all types [2-44], similarly, is actually a pressure driven longitudinal wave like the ordinary sound mode in neutral gas. In normal two-component plasmas, the electron thermal pressure drives the collective ion oscillations to propagate as the ion sound (acoustic) wave. Here the electron thermal pressure provides the restoring force to allow the collective ion dynamics in the form of ionic compression and rarefaction to propagate in the plasma background and ionic mass provides the corresponding inertial force. Thermal plasma species (like electrons) are free to carry out thermal screening of the electrostatic potential. In absence of any dissipative mechanism, the ion sound wave moves with constant amplitude. For mathematical description of the ion sound kinetics, the plasma electrons are normally treated as inertialess species and the plasma ions, with full inertial dynamics. However, recent finding of ion sound wave excitation in transonic plasma condition of hydrodynamic equilibrium offers a new physical scope of acoustic turbulence due to weak but finite electron inertial delay effect [5-12]. Qualitative and quantitative modifications are introduced into its nonlinear counterpart as well, under the same transonic plasma equilibrium configuration [12]. The transonic transition of the plasma flow motion quite naturally occurs in the neighborhood of boundary wall surface of laboratory plasmas, self-similar expansion of plasmas into vacuum, in solar wind plasmas and different astrophysical plasmas, etc. The self-similar plasma expansion model predicts supersonic motion of plasma flow into vacuum. This model is widely used to describe the motion of intense ion plasma jets produced by short time pulse laser interaction with solid target [17-

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23]. Recently, the self-similar plasma expansion into vacuum is modeled by an appropriate consideration of space charge separation effect on the expanding front [13]. According to the recently proposed inertia-induced ion acoustic excitation theory [5-8], the large-scale plasma flow motion feeds the energy to the short scale fluctuations near the pre-sheath termination at sonic point. This is a kind of energy transfer process from large-scale flow energy to wave energy through short scale instability of cascading type. In order to maintain the turbulence type of hydrodynamic equilibrium, there must be some source to feed large-scale flow and sink to arrest the infinite growth of the excited short waves. The growing wave energy could be used to re-modify the global transonic equilibrium such that the transonic transition becomes a natural equilibrium with smooth change in flow motion from subsonic to supersonic regime. Of course, this is a quite involved problem to handle the self-consistent turbulence theory of transonic plasma in terms of anomalous transport [5]. Now one may ask how to produce such boundary layer with sufficient size of the transonic plasma layer for laboratory experimentations? This, in fact, is an experimental challenge to design and set up such experiments to produce

extended length of the transonic zone to sufficient extent to resolve the desired unstable

wave spectral components. Creation of a thick boundary layer of transonic flow dynamics

is, no doubt, an important task. This zone lies between subsonic and supersonic domains,

and is naturally bounded by low supersonic and high subsonic speeds. It should be

mentioned here that the sonic velocity corresponds to the phase velocity of the bulk

plasma mode of the dispersionless ion acoustic wave. In case of sheath edge boundary,

transonic layer could be probed by high-resolving diagnosis of the Debye length order.

The desired experiments of spectral analysis of the unstable ion acoustic waves in

transonic plasma condition may be quite useful to resolve the mystery of sheath edge

singularity. Using de-Lavel nozzle mechanism of hydrodynamic flow motion, experiments

could be designed to produce transonic transition layer of desired length and

characteristics [6-8].

Study of the ambient acoustic spectrum associated with plasma flow motion can be termed

as the acoustic spectroscopy of equilibrium homogeneous plasma flows [6, 26]. This may be

useful for expanding background plasmas [13], solar wind plasmas and also in space

plasmas through which the space vehicles’ motion and aerodynamic motion occur [3, 25,

28]. Basic principles of the acoustic spectroscopy have concern to the linear and non-linear

ion acoustic wave turbulence theory and properties of the transonic plasma equilibrium [5-

12, 26]. These properties may be used to develop the required diagnostic tools to study and

describe the hydrodynamic equilibrium states of plasma flows by suitable observations and

analysis of the waves and instabilities they exhibit. In fact, the ambient turbulence-driven

plasma flow is quite natural to occur in toroidal and poloidal directions of the magnetic

confinement of tokamak device. Similar physical mechanism is supposed to be operative in

the transonic transition behavior of equilibrium plasma flow motion [5-12]. Thorough

investigations of acoustic wave turbulence theory in transonic plasma condition will be

needed to explore transonic flow dynamics on a concrete footing.

Recently, there has been an outburst of interest in plasma states where the assumption of

static equilibrium practically is violated [28-30]. Great deals of research activities are now

going on in transonic and supersonic magnetohydrodynamic (MHD) flows in laboratory

and astrophysical plasmas. Similar activities are also important for understanding the

designing of supersonic aerodynamics having relevance in spacecraft-based laboratory

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experimentations of space plasma research as well [8, 30]. This is also argued that future

tokamak reactors need the consideration of rotation of fusion plasma with high speeds that do

not permit the assumption of static equilibrium to hold good. This may be brought about due

to neutral beam heating and pumped divertor action for the extraction of heat and exhaust. In astrophysics [3, 28-32, 35-44], the primary importance of plasma flows is revealed in such diverse situations as coronal flux tubes, stellar winds, rotating accretion disks, torsional modes, and jets emitted from radio galaxies. This is to argue that the basic understanding of the acoustic wave dynamics in transonic plasma system constitutes an important subject of future interdisciplinary research [5-12, 26-30]. This may be useful for development of the appropriate diagnostics for acoustic spectroscopy to measure and characterize the hydrodynamic equilibrium of flowing transonic plasmas [8-10]. Such concepts of acoustic wave dynamics in a wider horizon may also applied to understand some helio- and astero-seismic observations in astrophysical contexts. Most of the plasma devices of industrial applications like dense plasma focus machine, plasma torches, etc. depend on the plasma flows that violate the static equilibrium [26-30]. In fusion plasmas of future generation too, the static approximation of the equilibrium plasma description may not be suitable to describe the acoustic wave behavior. In future course of fusion research, rotational motions of fusion plasmas in poloidal and toroidal directions may decide the equilibrium. This is important to state that in toroidal plasmas, the geodesic acoustic mode becomes of fundamental importance in comparison to the ordinary sound modes [30]. This may be more important when these rotational motions are in the defined range of the transonic limit. Simplicity is correlated to the local mode approximation of the acoustic wave description in transonic limit of uniform and unidirectional plasma flow motion without magnetic field. The lowest order nonlinear wave theory of the ion acoustic wave dynamics predicts that the usual KdV equation is not suitable to describe the kinetics of the nonlinear traveling ion acoustic waves in transonic plasma condition [8-9, 12, 26]. A self-consistent linear source driven KdV equation, termed as d-KdV equation, is prescribed as a more suitable nonlinear differential equation to describe the nonlinear traveling ion acoustic wave dynamics in transonic plasma condition. By mathematical structure of the derived d-KdV equation, it looks analytically non-integrable and physically non-conservative dynamical system [8-9]. Due to linear source term, an additional class of nonlinear traveling wave solution of oscillatory shock-like nature is obtained. This is more prominent in the shorter scale domain of the unstable ion acoustic wave spectrum, but within the validity limit of weak nonlinearity and weak dispersion. If there is multispecies ionic composition in a plasma system, varieties of plasma sound waves are likely to exist depending on, in principle, the number of inertial ionic species. In plasmas containing two varieties of dust or fine suspended particles, two distinct kinds of natural plasma sound modes are possible [15-16]. Such plasmas, termed as the colloidal plasmas [16], have become the subject of intensive study in various fields of physics and engineering such as in space, astrophysics, plasma physics, plasma-aided manufacturing technique, and lastly, fusion technology [14-23]. The dust grains or the solid fine particles suspended in low temperature gaseous plasmas are usually negatively charged. It is also observed that plasmas including micro scale-sized and nano scale-sized suspended particles exist in many natural conditions of technological values. Such plasmas have been generated in laboratories with a view to investigate the dust grain charging physics, plasma wave physics as well as some acoustic instability phenomena.

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The two distinct sound modes, however, in bi-ion colloidal plasma are well-separated in space and time scales due to wide range variations of mass scaling of the normal ions and the charged dust grains and free electrons’ populations. The charged dust grains are termed as the Dust Grain Like Impurity Ions (DGLIIs) [15] to distinguish from the normal impurity ions. The present contributory chapter, additionally, applies the inertia-induced acoustic excitation theory to nonlinear description of plasma sound modes in colloidal plasma [15-16] under different configurations. Two separate cases of ion flow motion and dust grain motions are considered. It is indeed found that the modified Ion Acoustic Wave (m-IAW) or Dust Ion Acoustic (DIA) wave and the so-called (Ion) Acoustic Wave (s-IAW) or Dust Acoustic (DA) wave both become nonlinearly unstable due to an active role of weak but finite inertial correction of the respective plasma thermal species [15, 26]. Proper mass domain scaling of the dust grains for acoustic instability to occur is estimated to be equal to that of the asymptotic mass ratio of plasma electron to ion as the lowest order inertial correction of background plasma thermal species. This contributory chapter is thus a review organized to aim at some illustrative examples of linear and nonlinear acoustic wave propagation dynamics through transonic plasma fluid, particularly, under the light of current scenario. Some important reported findings on nonlinear acoustic modes found in space and astrophysical situation [31-44], like in solar plasma system [10-11, 31-44], will also be presented in concise to understand space phenomena. Incipient future scopes of the presented contribution on transonic flow dynamics in different astrophysical situations will also be briefly pointed out.

2. Physical model description

A simple two-component non-isothermal, field-free and collisionless plasma system under fluid limit approximation is assumed. The plasma ions are supposed to be drifting with uniform velocity at around the sonic phase speed under field-free approximation. Global plasma equilibrium flow motion over transonic plasma scale length at hydrodynamic equilibrium is assumed to satisfy the global quasi-neutrality. Such situations are realizable in the transonic region of the plasma sheath system as well as in solar and other stellar wind plasmas [3, 10-11, 35-41]. Its importance has previously been discussed [1, 5-12], where the ion-beam driven wave phenomena are supposed to be involved in Q-machine or in unipolar/ bipolar ion rich sheath formed around an electrode wall or grid in Double Plasma Device (DPD) experiments of plasma sheath driven low frequency instabilities of relaxation type [5]. The unstable situation is equally likely to occur on both the sides of the sheath structures with plasma ion streamers [12].

3. Linear normal acoustic mode analyses

3.1 Basic governing equations The basic set of governing dynamical evolution equations for the linear normal mode behavior of fluid acoustic wave consist of electron continuity equation, electron momentum equation, ion continuity equation, and ion momentum equation [5-6]. The set is closed by coupling the plasma thermal electron dynamics with that of plasma inertial ion dynamics through a single Poisson’s equation for electrostatic potential distribution due to localized ambipolar effects. Applying Fourier’s wave analysis for linear normal mode behavior of ion acoustic wave over the basic set of governing dynamical evolution equations [5-6], the linear dispersion relation is derived as follows

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( )( )

( )te D e ak v k

k v

2 2 2 22 2 2

0 2

1. .

λ+Ω + = Ω − Ω

Ω (1)

All the notations in the equation (1) are usual and conventional. Here Ω is the Doppler-

shifted frequency of the ion acoustic wave, aΩ is the ion acoustic wave frequency in laboratory frame of reference, k is the angular wave number of the ion acoustic wave such

that kλDe is a measure of the acoustic wave dispersion scaling and vte is the electron thermal velocity. Now the kinematics of any mode can be analyzed in two different ways: one in lab-

frame and the other, in Doppler-shifted frame of reference. This is to note that the obtained

dispersion relation differs from those of the other known normal modes of low frequency

relaxation type of instability, ion plasma oscillations and waves. This is due to the weak but

finite electron inertial delay effect in the dispersion relation of the wave fluctuations. This is

mathematically incorporated by a weak inertial perturbation over electron inertial dynamics

over the leading order solution obtained by virtue of electron fluid equations neglecting

electron inertial term.

It is thus obvious from the mathematical construct of equation (1) that the LHS is a non-

resonant term whereas RHS is a resonant term. The RHS gets artificially transformed into a

resonant term if and only if ik v 0. 0< . Now, it can be inferred that equation (1) represents a resonantly unstable situation at Doppler shifted resonance frequency of ak v0.Ω ≈ ≥ Ω , if

and only if k v0.

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It is clear from the equation (3) that two poles are possible to exist in Ω -space at 0Ω = and

k v0.Ω = for k v0. 0< . According to graphical method, the beam-plasma system will exhibit instability only when the curve of ( )F k,Ω versus Ω has multiple singular values in Ω -space having finite minima in between the two successive singularities, which do not

intersect with the line ( ) ( )DeF k k2 2, 1 1 λΩ = + . The required condition for minimization of ( )F k,Ω in Ω -space can be obtained by equating dF d 0Ω = . Now this condition, when

applied to equation (3), results into the following equality to derive the value of Ω where

dispersion function is supposed to be minimum

( ) ( )a ak v2 2 20. 0.Ω Ω + + Ω Ω − Ω = (4)

In principle, equation (4) is to be solved to determine the value of Ω . This is obvious to note

that this equality is satisfied at resonance value of ak v0~ . ~Ω Ω for k v0. 0< . Now to indemnify the complex nature of Ω , the functional value of ( ) ( )DeF k k2 2, 1 1 λΩ > + . This can, however, be further simplified to yield the following inequality to determine the threshold

value for the onset of the inertia-induced instability

( ) ( ) ( )te a Dek v k v k22 2 2 2 2 2

0. 1 .λΩ − Ω > Ω − + (5)

The threshold condition for the instability is satisfied for equality sign at resonance

frequency ak v0~ . ~Ω Ω that characterizes the case of a marginal instability. A few typical

plots of the function ( )F k,Ω in Ω -space for shorter and longer acoustic wavelengths (perturbation scale lengths) are represented in Fig. 1.

3.3 Numerical analysis

Numerical techniques for solving polynomials over years have developed to a vast extent

for solving polynomials even with complex coefficients and complex variables. For the

present case, the Laguerre's algebraic root-finding method [6] to solve the normalized form

of polynomial equation has been used. The polynomial ( )P 'Ω in the normalized form of the dispersion relation (1) in ion-beam frame is given below

( )P a a a a a' ' '2 '3 '40 1 2 3 4 0.Ω = + Ω + Ω + Ω + Ω = (6)

Here all the normalized notations used are usual, generic and defined by

pi a a pi De te te s i ek k v v c m m' ' ', , 'ω ω λΩ = Ω Ω = Ω = = = and sM v c0= . The expressions for

the various coefficients in the polynomial ( )P 'Ω are defined as follows

( )( )te aa k k v'2 '2 '2 '20 1= − + Ω ,

a1 0= ,

( ) ( )( )tea k M k k v2 2 2 2

2 '. 1 ' ' '= + + ,

a k M3 2 '.= − , and

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a4 1= .

It is found that out of four possible roots of ( )P 'Ω , only two roots are complex and these are the complex conjugates as a pair. For all the complex conjugated roots, only the complex

root with positive imaginary part is useful, since this determines the growth rate of the

instability. Real and imaginary parts of the corresponding complex roots are then plotted as

shown in Figs. 2 and 3, respectively. Numerical characterization of the unstable mode of the

instability clearly depicts the resonant character of the electron inertia-induced resonant

acoustic instability [5].

3.4 Evaluation of wave energy

This is important to evaluate the wave energy in order to have a more complete picture of

the basic source mechanism of the discussed instability. In presence of the beam, it is

expected that one of the modes involved, has positive energy and the other has negative

energy. The dispersion relation (1) can be put in the laboratory frame for a more clear

identification and characterization of the positive and negative energy modes in the form of

dispersion function ( )k,ε ω as follows

( )( )

pe pi

De te

kk k v k v

2 2

22 2 2 2

0

1, 1 1 0.

.

ω ωε ω

λ ω

= + + − = − (7)

The average electric field energy stored in a propagating electrostatic (created by ambipolar effect) wave in a medium is given by the following relation [6]

( ) ( ) ( )W k E k k2

0

1, , ,

2ω ω ε δ ω ωε ωω

∂= ∂ . (8)

Here 0ε is the dielectric constant of free space, ( )E k,δ ω is the electric field amplitude of the ion acoustic fluctuations and ( )EW E k

21 2 ,δ ω= is the corresponding counterpart of electric

energy of the acoustic fluctuations through free space. Applying the equations (7) and (8),

the following can explicitly be derived

( )

( )pi

E De te

kW

W k k v k v

22

32 2 2 2

0

2, 2.

.

ωωωε ω ω

ωω λ ω

∂= = +

∂ − (9) Now, clearly, it is evident that the second term of equation (9) contributes negative energy value to the defined wave-plasma system. This occurs as because the sign of this term

becomes negative for the values of k v0.ω < , which is the case for the reported instability.

From a few typical plots in Fig.4, one can notice that the total wave energy suffers a sharp transition from negative to positive values at resonance frequency point of zero energy value. The resonance point lies in the domain of near-zero and non-zero frequencies in lab-frame. According to conventional definition and understanding, the wave energy expression in equation (9) classifies the near-zero frequency mode as the negative energy mode. Then immediately the non-zero frequency mode may be classified as the positive energy mode.

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This is important to clarify that the theoretical concept of near-zero frequency mode is an outcome of the mathematical construct of weak but finite electron inertial response to the ion acoustic wave fluctuations. The blowing up character, as shown in Fig. 4, of the total wave energy in opposite directions suggests referring the discussed instability to as an 'explosive instability' in accordance with the law of conservation of energy. It signifies the transonic plasma condition with the resonant mode-mode coupling of the positive and negative energy modes. The time average of the hydrodynamic and wave potential energies of the considered wave-plasma system over the growth time scale is conserved during the energy exchange process between the unstable resonant eigen modes and the main source of ion flow dynamics. These two modes are clearly identified from equation (9) as the natural resonant modes of the defined plasma system that undergo linear resonant mode-mode coupling to produce the defined wave instability.

3.5 Estimation of quenching time

Under the cold ion approximation, even the small electrostatic potential will be able to

distort the ion particle motion and associated trajectories, affecting the driving source flow

velocity of the resonant instability under consideration. In wave frame, the streaming ion

energy ( )iE can be expressed by the following relation

i iE m vk

2

0

1

2

ω = − . (10)

For ( )o sv k c v0~ω>> − , which is a valid case for the considered instability [5], the condition for ion orbit distortion becomes of the following form,

w iW m n v2

0 0

1

2≥ . (11)

From this condition, the quenching time is estimated under the assumption that the wave amplitude grows sufficiently from thermal noise level to physically measurable level such that

( ) tE iW t W eγ= . (12)

Here iW is the initial energy of the acoustic wave amplitude, which is of the order of the thermal fluctuations, i.e., i e DeW T

3~ λ and is the unnormalized linear growth rate. Using the resonance values of sk c v0ω = − and sk v kc0. ~ω − as derived in [5] for long wavelength case of resonant mode, equation (12) for the quenching time τ with the help of (9) can be rewritten as follows

( ) ( )( )e De Dei De

m MM n k

m k M

21 2 3 2 2

0

1 11 ln

2 4 1τ λ λ

λ

− = −

− . (13) For some typical plasma parameters in hydrogen plasma, ( )Den 30ln ~ 15 30λ − . For

Dek ~ 0.3, 0.1, 0.05λ near resonant M as in Figs. 2 and 3, equation (13) gives 1τ > , i.e., q piτ τ>. This physically means that the resonant growth time scale is greater than that of the plasma

ion oscillation time scale. Thus the resonant nature of the instability is observable in the

present analysis.

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3.6 Physical consequences

Wave energy analyses are carried out to depict the graphical appearance of poles (Fig. 1), nature of real parts of the roots (Fig. 2), nature of imaginary parts of roots (Fig. 3) and positive-negative energy modes (Fig. 4).

Fig. 1. Graphical appearance of resonance poles as a variation of the dispersion function

( )F k,Ω with normalized Doppler-shifted frequency for dispersion scaling (a) Dek 0.3λ = , (b) Dek 0.1λ = , and (c) Dek 0.01λ =

Fig. 2. Variation of the real part of the normalized Doppler shifted eigen mode frequency

( )′Ω with respect to Mach number ( )M for different values of 0.30,0.10,0.05Dekλ =

It is found that the instability arises out of linear resonance mode-mode intermixed coupling between the negative and positive energy modes. The total energy of the coupled mode-mode system comprising of hydrodynamical potential energy and wave kinetic energy, however, is in accordance with the law of conservation of energy in the observation time scale on the order of ion acoustic wave time scale. Identification and characterization of the resonance nature of the said instability through transonic plasma is presented in order to explore the acoustic richness in terms of collective waves, oscillations and fluctuations. This is an important point to be mentioned here that the same type of instability features are

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expected to happen in plasma-wall interaction process and sheath-induced instability phenomena in other similar situations as well. There are different sorts of analytical and numerical tools for studying the linear instabilities

in a given plasma system. Energy method, based on energy minimization principle and the

normal mode analysis, based on equilibrium perturbations are the two basic mathematical

tools for analyzing the stability behavior of the given plasma systems. However, the latter is

most popular and simple for common use in analyzing the threshold conditions of the

instabilities and their growth rates. In the normal mode analysis, a linear dispersion relation

is derived which can be put in the form of a polynomial with real or imaginary coefficients.

The limitation of the analytical method depends upon the degree of the polynomial.

Computational technique broadly takes into account two ways of investigating instability.

First, an unstable mode can be deduced by the derived dispersion relation. The obtained

polynomial is then solved to delineate the complex roots having concern to the desired

instability. Second, a more comprehensive computational method involves solving for the

time dependent solution. Simulation technique used to solve the basic set of equations is

supposed to give more complete picture of the space and time evolution of the wave

phenomena. However, there is another very informative and simple method for analyzing

the derived dispersion relation to predict for the unstable behavior of the plasma system

under consideration. This is the graphical method in which the dispersion relation is

graphically represented for different values of resonance characterization parameters.

Source perturbation scale length ( )Dekλ and deviation from sonic point ( )M1 − are the characterization parameters for the defined acoustic resonance.

Fig. 3. Variation of the normalized growth rate of the electron inertia-induced resonant

acoustic instability with Mach number for (a) 0.3Dekλ = , (b) 0.1Dekλ = and (c) 0.05Dekλ =

showing that transonic plasma is rich in wide range acoustic spectral components and

hence, an unstable zone

This is quite natural and interesting to argue that the transonic plasma condition offers a unique example where the physical situation of localized hydrodynamic equilibrium of quasi-neutral plasma flow dynamics exists. Previous publication reports that the transonic plasma layer, assumed to have finite extension, can be considered as a good physical situation to study the acoustic instability, wave and turbulence driven by electron inertia-induced ion acoustic excitation physics.

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Fig. 4. Explosive nature of the electron inertia-induced ion acoustic wave instability as an

outcome of an interplay for the linear resonant mode-mode coupling of positive and

negative energy eigen modes. It shows how the normalized wave energy varies with

normalized frequency under a set of fixed values of Mand Dekλ as (a) 0.85, 0.30DeM kλ= = ;

(b) 0.842, 0.100DeM kλ= = ; and (c) 1.79, 0.01DeM kλ= =

In the present sections of the chapter, many features about the electron inertia-induced ion acoustic wave instability are observed. For example, we physically identify and demonstrate the following features of the instability obtained by theoretical and numerical means of analysis of the desired dispersion relation: 1. The transonic plasma layer is an unstable zone of hydrodynamic equilibrium of

quasineutral plasma gas flow motion, 2. The instability is an outcome of the linear resonant mode-mode coupling of positive

and negative energy modes, 3. The quenching time of the instability is estimated for some typical values of plasma and

wave parameters as mentioned in the previous section. It is found to moderately exceed the ion plasma oscillation time scale, and

4. Lastly, this indicates that in lab frame observation the unstable mode of ion acoustic wave fluctuations at reduced frequencies may look like a purely growing mode. This is very likely to occur for almost entire unstable frequency domain of the frequency transformed ion acoustic waves.

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In fact, the electron inertial responses naturally appear only at electron oscillation frequency. However, the transonic plasma condition creates a natural physical situation for the same to occur even at the ion acoustic wave frequency of the transformed reduced values. The linear process of resonant mode-mode coupling produces this and makes the coupled system of wave modes unstable. We have identified and demonstrated the following features of the instability obtained by theoretical and numerical analysis of the dispersion relation: (i) The transonic plasma layer is indeed an unstable zone of hydrodynamic equilibrium of quasi-neutral plasma gas flow motion. (ii) The instability is an outcome of linear resonant mode-mode coupling of positive and negative energy modes. (iii) The normalized values of Doppler-shifted resonant frequencies of the unstable ion acoustic wave fluctuations in ion beam frame come out to be almost equal to 0.5. (iv) The estimated quenching time of the instability exceeds the ion plasma oscillation time scale moderately and hence, (v) In the lab-frame, the unstable modes of ion acoustic wave fluctuations at reduced frequencies may look physically like a purely growing mode. This is further argued that the physical insights as listed above can be useful as theoretical, graphical and numerical recipes to (1) formulate and solve the problems of saturation mechanisms of the unstable ion acoustic wave fluctuations, (2) formulate and solve the problems of the ion acoustic wave turbulence, and (3) design and setup experiments to study the basic physics of linear and nonlinear ion acoustic wave activities in unique transonic plasma system. These investigations may be useful to improve the existing conceptual framework of physical and mathematical methods of two-scale theory of plasma sheath research to resolve the long-term mystery of the sheath edge singularity. These, in brief, are added to judge the didactic vis-à-vis the scientific qualities of the current research work too much specialized in the subject of ion acoustic wave physics.

3.7 Comments

The main conclusive comment here is that the graphical method successfully explains the unstable behavior of the fluid acoustic mode of the ion acoustic wave fluctuations in drifting plasmas with cold ions and hot electrons. A more vivid picture of linear resonant mode-mode coupling of positive and negative energy waves is obtained. This is important to note that simple formulae for wave energy and quenching time calculations [6] are used. This calculation further confirms the earlier results of stability analysis of drifting plasmas against the acoustic wave perturbations [5]. It is, therefore, reasonable to think of logical hypothesis of wave turbulence model approach to solve the sheath edge singularity problem [1, 4]. Actually, the local normal mode theory of the discussed instability implies that the entire transonic plasma zone should be rich in wide frequency range spectrum of the ion acoustic wave fluctuations. This leads to develop the conceptual framework of situational definition of the Debye sheath edge to behave as a turbulent zone with finite extension [12]. This hypothetical scenario of the transonic plasma condition can be examined by appropriate experiments of measuring wide range spectral components of the ion acoustic wave fluctuations. This is a nontrivial problem to explicitly characterize the turbulent properties of the transonic region. The more realistic problem of wave turbulence analysis demands the self-consistent consideration of flow induced quasi-neutral plasma with inhomogeneity in equilibrium plasma background. Similar situations are likely to occur in stellar wind plasmas, where, the transonic behavior is brought about by deLaval nozzle mechanism [6-10]

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of gas flow through a tube of varying cross section. Recent experimental observation [12] in double plasma device (DPD) reports an instability even in a condition of symmetric bipotential ion-rich sheath case. Its frequency falls within zero frequency range and its source is believed to lie in presheath. Finally, in a nutshell, it is concluded that the graphical method of analyzing the dispersion relation of the inertia-induced instability offers a simple and more informative method of practical importance in transonic plasma equilibrium. Moreover, the plasma environment of Debye sheath edge locality offers a realistic situation for self-excitation of the ion acoustic wave turbulence through resonant ion acoustic wave instability. This is induced by hydrodynamic tailoring of the ion acoustic wave-induced electron density fluctuations. Of course, no experimental observation of instability in transonic plasma has yet been reported to directly compare with the theoretical results. However it cannot be undermined in understanding wave turbulence phenomenon of flowing plasmas. This is informative to add that the frequency and amplitude transformation of the normal ion acoustic wave into unstable ion plasma wave at higher frequency is reported in high intense laser–plasma interaction processes [6-7] through the nonlinear ponderomotive action. This leads to the formation of soliton, double layers, etc. through the saturation mechanism of strong laser-plasma interaction processes due to non-zero average value of the spatially varying electric field associated with laser pulse.

4. Nonlinear normal acoustic mode analyses

4.1 Basic governing equations

A large amount of literature of theoretical and experimental investigations has been produced on the solitary wave propagation in plasmas since the theoretical discovery of ion acoustic soliton [4, 11-12, their references]. Varieties of physical situations of drifting ions of high energy with [5-12] and without [13-33] electron inertial correction have been considered in the ion acoustic wave dynamics. It is shown that the electron inertial motion becomes more important than the ion relativistic effect. Such situations exist in Earth's magnetosphere, stellar atmosphere and in Van Allen radiation belts [3]. Similar studies have been carried out in plasmas with additional ion beam fluid with full electron inertial response in motion [12 and references]. A number of experiments were performed in the unstable condition of beam plasma system in laboratory in order to observe soliton amplification [12]. There are many theoretical calculations and experiments on linear [7-8] and nonlinear [9-11] wave propagation properties of acoustic waves to see their behavior near the transonic point. For an assumed transonic region, it has been theoretically shown that the small amplitude acoustic wave fluctuations exhibit linear resonant growth of relaxation type under the consideration of weak but finite electron inertial delay effect [12-13]. In contrast to earlier claim [3] that the complex nature of coefficients in KdV equation prevents the soliton formation, we argue that their interpretation seems to be physically inappropriate. Instead, by global phase modification technique [12], we show that the usual soliton solution exists (even under the unstable condition), but only for infinitely long wavelength source perturbations. Otherwise, oscillatory shock-like solutions are more likely to exist. Under fluid approximation, the self-consistently closed set of basic dynamical equations for transonic plasma system with all usual notations in normalized form is given as follows

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Electron continuity equation:

eev

vt x x

. 0φ φ∂ ∂ ∂

+ + =∂ ∂ ∂

, and (14)

Electron momentum equation:

e e e eei e

m v v nv

m t x x n x

1.

φ∂ ∂ ∂ ∂ + = −

∂ ∂ ∂ ∂ . (15) This is to remind the readers that equation (15) is obtained by substituting zero-order

solution of Boltzmann electron density distribution into the normal electron continuity

equation. In fact, in the asymptotic limit of e im m 0→ , electron continuity equation as such is redundant as because the left hand side (electron inertial effect) of (15) is ignorable.

Equation (14) basically offers a scope to introduce the weak but finite role of electron to ion

inertial mass ratio on the normal mode behavior of acoustic wave. Ion continuity equation:

( )i i in

n vt x

0,∂ ∂

+ =∂ ∂

(16)

Ion momentum equation:

i iiv v

vt x x

φ∂ ∂ ∂+ = −

∂ ∂ ∂, and (17)

Poisson equation:

e in nx

2

2

φ∂= −

∂. (18)

Following form of the derived d-KdV equation obtained from the above equations by the

standard methodology of reductive perturbation [12] describes the nonlinear ion acoustic

wave dynamics under transient limit (~soliton transit time scale) in a new space defined by

the stretched coordinates ( ),ξ τ . This is to mention that ( ) ( )x t e, , γτφ φ ξ τ −= and 0γτ → under the transient time action of the propagating ion acoustic soliton through transonic

plasma

K M Kt x x

3

0 0 03

1

2

φ φ φφ γ φ

∂ ∂ ∂+ + =

∂ ∂ ∂. (19)

Here the notations K0 and M0 termed as complex response coefficients [11-12, 26] and the linear

resonant growth rate (γ) of the ion acoustic wave with complex Doppler-shifted Mach number D Dr DiM M iM= + and lab-frame Mach number r iM M iM= + in transonic equilibrium appearing in equation (19) are as follows,

K A B1 22 2

0 = + where,

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( )r Dr Dr i

m Dr i

M M M MA

M M

3 2

32 2

3

ε

− = + +

, and

( )i i Dr i

m Dr i

M M M MB

M M

3 2

32 2

3

ε

− = + +

,

M C D1 22 2

0 = + where,

( ){ }( )

( )Dr i Dr i r i r imDr i

M M M M M M M MC

M M

22 2 2 2 22 2 2 2

4 22 2

3 4 411

2 ε

− − − − = − −

+ , and

( )( )

( )Dr i Dr i r i r imDr i

M M M M M M M MD

M M

2 2 2 2

4 22 2

12 41

2 ε

− − = − +

+ ,

( )i De ie

mk v

m

1 2

02 1γ λ

= − . The notations are usual and generic as discussed earlier [12]. In the system, plasma ions are self consistently drifting or streaming through a negative neutralizing background of hot electrons having relatively zero inertia. The time response of the electron fluid here is normally ignored. As a result, the unique role of weak but finite electron inertia to destabilize the plasma ion sound wave in transonic plasma equilibrium even within fluid model approach of normal mode description is masked.

4.2 Physical consequences

Now equation (19) after being transformed into an equivalent stationary ODE form by the

Galilean transformations is numerically solved as an initial value problem. Some very small

simultaneous values of φ , φ ξ∂ ∂ and 2 2φ ξ∂ ∂ are required for the numerical programme to proceed. A few numerical plots for the desired nonlinear evolutions are shown in Figs. 5-

6. This is to note that the calculated amplitudes (as shown in Figs. 5a-6a) are the solutions of

the present d-KdV equation (19) with bounded and unbounded phase potraits (as shown in

Figs. 5b-6b). Now, the actual amplitudes of the resulting solutions can be deduced by

multiplying the numerically obtained values with ( )DEk2 2~ 10ε λ −≈ [12]. In principle, the

parameter ε is an arbitrary smallness parameter proportional to the dispersion strength or the amplitude of the weakly dispersive and weakly nonlinear plasma wave. The unique motivation here is to characterize the possible nonlinear normal mode structure of ion acoustic fluctuations under unstable condition of the ion drifts [8-9,12]. By this very specific example, we show that the complex nature of the coefficients of the derived KdV equation in the unstable zone of transonic plasma doesn't prevent the existence of localized nonlinear solutions including usual soliton solution, too. The concept of global phase

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modification technique (DPMT) [11-12, 26] results into a d-KdV equation [8-9, 12] with variable nonlinear and dispersion coefficients. Two distinct classes of solutions are obtained: soliton and oscillatory shock-like structures.

Amplification and damping of the driven KdV soliton over the usual KdV soliton is noted

for extremely large wavelength (dc) acoustic driving in source term as shown in Figs. 5-6.

The amplification near resonance is associated with considerable reduction in nonlinear

coefficient than unity as confirmed by numerical calculation. In other cases of shorter

acoustic driving in source term as shown in Figs. 5-6, nonlinear solutions of oscillatory

shock-like nature are obtained depending on the small deviation from resonant values. It is

clearly seen that the peaks of oscillatory shock-like solutions are of either sinusoidal or non-

sinusoidal nature with continuous elevation of the initial values of the successive peaks

beyond the main nonlinear acoustic peak.

Most of the experimental results in Double Plasma Device (DPD) are reported to show that

the obtained theoretical results may have practical relevance to understand the basic physics

of ion acoustic wave activities in the transonic region [12] as in Fig. 7. The experiment is

performed in a DPD of 90 cm in length and 50 cm in diameter equipped with multi-dipole

magnets for surface plasma confinement [12]. The chamber is divided into source and target

by a mesh grid of 85% transparency kept electrically floating. It is evacuated down to a

pressure of ( ) 55 6 10−− × Pa with a turbomolecular pump backed by a rotary pump. Ar-gas is bled into the system at a pressure ( ) 23 5 10−− × Pa under continuous pumping condition. The source and target plasmas are produced by dc discharge between the tungsten filament

of 0.1 mm diameter and magnetic cages.

(a) (b)

Fig. 5. Profile of (a) ion acoustic potential ( )φ with normalized space variable ( )ξ , and (b) phase space geometry of ion acoustic potential in a phase space described by φ and ( )ξφ with 82.5 10Dekλ

−= × (fixed) for Case (1): 71.0 10 ,δ −= × Case (2): 72.5 10 ,δ −= × Case (3): 75.0 10 ,δ −= × and Case (4): 77.5 10δ −= ×

The plasma parameters are measured with the help of a plane Langmuir probe of 5 mm

diameter and Retarding Potential Analyzer (RPA) of 2.2 cm in diameter. The probe and the

analyzer are movable axially by a motor driving system so as to take data at any desired

-15 -10 -5 0 5 10 150

0.5

1

1.5

2

2.5

3Ion acoustic potential profile in coordination space

Normalized position

Ion a

coustic p

ote

ntial

Case (1)

Case (2)

Case (3)

Case (4)

0 0.5 1 1.5 2 2.5 3-1.5

-1

-0.5

0

0.5

1

1.5Ion acoustic potential profile in phase space

Ion acoustic potential

Ion a

coustic p

ote

ntial gra

die

nt

Case (1)

Case (2)

Case (3)

Case (4)

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position. The plasma parameters are: electron density n cm8 9 30 10 10= − , electron

temperature eT eV1.0 1.5= − and ion temperature iT eV0.1= . An ion-acoustic wave is excited with a positive ramp voltage of which the rise time is controllable and is applied to

the source anode of the system. Propagating signals are detected by an axially movable

Langmuir probe which is biased to V4+ with respect to the plasma potential in order to detect the perturbation in the electron current saturation region. The current is then

converted into voltage by a resistance of 100Ω and the resultant signals are fed to the oscilloscope. The probe surface is repeatedly cleaned with ion bombardment by applying

V100− to it for a short time scale.

(a) (b)

Fig. 6. Same as Fig. 5 but with 11.0 10Dekλ−= × (fixed) for Case (1): 51.0 10 ,δ −= × Case (2):

52.0 10 ,δ −= × Case (3): 53.0 10 ,δ −= × and Case (4): 55.0 10δ −= ×

.

Fig. 7. Experimental profiles of variation of plasma density perturbation ( )nδ against time ( )t at different position of the probe from the grid is shown. Along the x-axis, each division represents 10 sµ and along the y-axis, the density perturbation scale is given as 0.09en nδ =

-15 -10 -5 0 5 10 150

0.5

1

1.5

2

2.5Ion acoustic potential profile in coordination space

Normalized position

Ion a

coustic p

ote

ntial

Case (1)

Case (2)

Case (3)

Case (4)

0 0.5 1 1.5 2 2.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Ion acoustic potential profile in phase space

Ion acoustic potential

Ion

acou

stic

pote

nti

al g

radie

nt Case (1)

Case (2)

Case (3)

Case (4)

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It is also seen that the value of DrM1δ = − where resonance occurs remains invariant to spectral variation in source term even by orders of 1 101.0 10 1.0 10− −× − × . The nonlinear and dispersive coefficients exhibit very sensitive role on even slight variation of δ from its resonance. It is noted that as the value of 0δ = , the usual KdV soliton is recovered, irrespective of any wave number value in the source term. The source term plays an

effective role only when finite Dekλ and δ -values are assigned simultaneously

4.3 Comments As per experimental observations, oscillatory shocks are reported to emerge from the transonic zone in the target plasma as shown in Fig. 7. One can qualitatively argue that as soon as the solitary wave passes through the unstable transonic zone, it may experience the transient phase modifications leading thereby to the formation of oscillatory shock. The observed damping of the oscillatory shock may be correlated to the non-resonant type of dissipation through phase incoherence among ion acoustic spectral components of the usual solitary wave. It seems to be more plausible to argue that the input energy to the usual soliton due to transonic plasma equilibrium may be shared among different spectral components through adiabatic energy exchange processes. This is concluded here that the complex coefficients of the KdV equation should, in principle, not become the criterion for the non-existence of localized nonlinear solutions including usual solilton, too. But the usual soliton solution exists only for infinitely long wavelength source perturbation. This conclusion is derived subject to the validity condition of our arguments of global temporal phase modification of usual soliton amplitude under unstable condition of the plasma medium. The unstable condition of the medium may cause structural deformation of the non-driven KdV solution. Such deformations may result into sinusoidal (linear) or non-sinusoidal (nonlinear) peaks of oscillatory shock-like solution depending on the wavelength of the source perturbations [8-9, 12]. Applying the wave packet model for a moving soliton leaving behind an acoustic tail of dispersive waves known as precursor or acoustic wind (in soliton frame), the asymmetry can be associated with elevation of the bottom potential by a finite dc value superposed with periodic repetition of linear or nonlinear peaks. The amplification or suppression of a single soliton can be possible only for infinitely long wavelength (dc) source. For shorter wavelength source driving, the transition from usual soliton solution to oscillatory shock-like solutions is more likely to occur. It is, in brief, concluded that the present mathematical study of d-KdV equation offers a significant contribution of analytical supports to our numerical prediction of structural transformation of the traveling nonlinear ion acoustic waves in transonic plasma equilibrium of desired quality. It clearly shows that the actual solution of d-KdV equation is a resultant of linear mixing (superposition) of soliton and shock both. Dominating features of the individual nonlinear modes is decided by an appropriate choice of the specific values of unstable wave number (or wavelength) for a given value of the ion flow Mach number. It is obvious to note that in zero growth limit of d-KdV equation, the shock-term disappears and only soliton remains. This limit is correlated with dc range of the chosen unstable wave number of quite weaker dispersion strength. As the dispersion strength becomes significant to influence the original soliton strength of weak nonlineariy and weak dispersion in the defined transonic plasma of finite extension, structural modification of the usual KdV soliton profile occurs. We further argue that the linear and nonlinear normal mode behaviors of the ion acoustic waves in transonic plasma condition differ qualitatively from those derived for static and

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dynamic equilibriums without electron inertial correction. The finite but weak hydrodynamic tailoring of the electron fluid motion on ion acoustic space and time scales brings about this difference. It is then argued that the plasma flows in transonic equilibrium should exhibit rich spectrum of linear and nonlinear ion acoustic waves and oscillations. Of course, under Vlasov model the hot electrons with streaming velocity comparable to the phase speed of the ion sound wave may destabilize the ion sound mode through wave-particle resonance effect [8 and references] too. However, our excitation mechanism of ion sound wave differs from the other known mechanisms [8] to excite the same ion sound wave on many grounds [8]. This kind of theoretical scenario of transonic plasmas offers a unique scope of acoustic spectroscopy to describe the internal state of transonic equilibrium of plasma flows [28]. These calculations have potential applications [26] extensively to understand plasma acoustic dynamics in colloidal plasmas too, but under transonic equilibrium configuration. A generalized statement thereby is reported that all possible sound modes in multi-species colloidal plasmas with drift motions (of inertial ionic species) could be destabilized by the inertial delay effect of the corresponding plasma thermal species that carry out thermal screening of acoustic potential developed due to respective inertial ionic species. Of course, threshold values may differ depending on the choice of the plasma sound mode under consideration. In technological application point of view, one may argue that the proposed theoretical model for inertia-induced acoustic instability mechanism may be utilized to make a plasma-based micro device for acoustic amplifier [26]. The amplified acoustic signals (developed due to respective inertial ionic species) from the amplifier could be detected, received and analyzed for the diagnosis and characterization of hydrodynamic flow of plasmas with embedded inertial dust contaminations. These analyses may have potential applications in different ion acoustic wave turbulence-related situations like aerodynamics, solar wind and space plasmas, fusion plasmas, industrial plasmas and plasma flows in astrophysical context, etc.

5. Astrophysical normal acoustics

A plasma-based Gravito-Electorstatic Sheath (GES) model is proposed to discuss the fundamental issues of the solar interior plasma (SIP) and solar wind plasma (SWP). Basic concepts of plasma-wall interaction physics are invoked. Here the wall is defined by a continuous variation of gravity associated with the SIP mass. The neutral gas approximation of the inertially confined SIP is relaxed, and as such the scope of quasi-neutral plasma sheath formation is allowed to arise near the self-consistently defined solar surface boundary (SSB). Analytical and numerical results are obtained to define the SSB and discuss the physics of the surface properties of the Sun, and hence, those of the SWP.

5.1 Physical model description

The SIP system can be idealized as a self-gravitationally bounded quasi-neutral plasma with a spherically symmetric surface boundary of nonrigid and nonphysical nature. The self-gravitational potential barrier of the solar plasma mass distribution acts as an enclosure to confine this quasi-neutral plasma. An estimated typical value ~10-20 of the ratio of the solar plasma Debye length and Jeans length of the total solar mass justifies the quasi-neutral behavior of the solar plasma on both the bounded and unbounded scales. Here the zeroth-order boundary surface can be defined by the exact hydrostatic condition of gravito-electrostatic force balancing of the enclosed plasma mass at some arbitrary radial position

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from the center of the mean solar gravitational mass. With this much background in mind, let us now formulate the problem of the physical and mathematical descriptions GES formation around the SSB. For simplicity, we consider spherical symmetry of the inertially confined SIP mass, which helps to reduce the three dimensional problem of describing the GES into a simplified one dimensional problem in the radial direction. Thus, only a single radial degree of freedom is required for description of the dynamical behavior of the SWP under the assumed spherically symmetric self-gravitating solar plasma mass distribution.

The idea of the GES formation can be appreciated with quantitative estimates of the gravito-

thermal coupling constants for the SIP electrons and hydrogen ions. Henceforth, “ions” and

“hydrogen ions” will be used in the sense of the same ionic species. These parameters [10]

can be defined and estimated as follows: The gravitothermal coupling constant for electrons

can be estimated as e B e ek T m g R 10Θ ΘΓ = ≈ , for a mean electron temperature of eT5~ 10 K

and as e 800Γ ≈ for mean eT610= K. The notation Bk (=1.3806×10-23 JK-1) denotes the

Boltzmann constant. Similarly, the gravito-thermal coupling constant for ions can be

estimated as ( )i i e e i eT m T m 1Γ = Γ

K for the SIP emerges as a more suitable choice for our theoretical consideration.

According to our GES-model analyses, the GES divides into two scales: one bounded, and

the other unbounded. The former includes the steady state equilibrium description of the

SIP dynamics bounded by the solar self-gravity. This extends from the solar center to the

self-consistently defined and specified SSB. The unbounded scale encompasses the SWP

dynamics extending from the SSB to infinity. The SIP electrons can easily escape from the

defined SSB. On the other hand, the SIP ions cannot cross the gravitational potential barrier

of the solar mass on their thermal energy alone. However, surface leakage of the SIP

electrons is bound to produce an electrostatic field by virtue of surface charge polarization.

This, in turn, provides an additional source to act on the SIP ions to further energize and

encourage them cross over the solar self-gravitational potential barrier.

5.2 Basic governing equations

In order to describe the plasma-based GES physics of our model system, we adopt a

collisionless unmagnetized plasma fluid for simplicity in mathematical development to

obtain some physical insight into the solar wind physics. The role of magnetic field is also

ignored (just for mathematical simplicity) in discussing the collisionless SIP and SWP

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dynamics. Applying the spherical capacitor charging model [3], the coulomb charge on the

SSB comes out to be SSBQ C~ 120 . The mean rotational frequency of the SSB about the centre of the SIP system is is determined to be SSBf Hz

12~ 1.59 10−× [42]. Applying the electrical

model [42] of the Sun, the mean value of the strength of the solar magnetic field at the SSB in

our model analysis is estimated as SSB SSB SSBB Q f T2 114 ~ 7.53 10π −= × , which is negligibly

small for producing any significant effects on the dynamics of the solar plasma particles.

Thus the effects of the magnetic field are not realized by the solar plasma particles due to the

weak Lorentz force, which is now estimated to be ( )LF e v B N333.61 10−= × ≈ × corresponding

to a subsonic flow speed v cm s 1~ 3.00 − with the input data available [2, 42] with us and

hence, neglected. Therefore our unmagnetized plasma approximation is well justified in our

model configuration. In addition, the effects of solar rotation, viscosity, non-thermal energy.

For further simplification, the electrons are assumed to obey a Maxwellian velocity

distribution. Although these approximations may not be realistic, but they may be

considered working hypotheses to begin with an ideal situation. Deviations indeed exist

from a Maxwellian velocity distribution. We however use it as a working hypothesis for our

model considerations. As a result, the usual form of the Boltzmann density distribution for

plasma thermal electrons with all usual notations is given as

eN e .θ= (20)

Here e eN n n0= denotes the normalized electron density. The generic notation ee Tθ φ= denotes the normalized value of the plasma potential associated with the GES on the

bounded scale and with the SWP on unbounded scale. The general notation en stands for the nonnormalized electron density and in m0 ρΘ= defines the average bulk density of the equilibrium SIP. The notation 1.43ρΘ = g cm-3 stands for the average but constant solar plasma mass density and im

241.67 10−= × g for the ionic (protonic) mass. Again e

represents the electronic charge unit and φ , the nonnormalized plasma potential associated with both the GES and SWP. The hydrogen ions are described by their full inertial response dynamics. This includes the ion momentum equation as well as the ion continuity equation. The first describes the change in ion momentum under the action of central gravito-electrostatic fields of potential gradient and forces induced by thermal gas pressure gradients. The latter equation is considered a gas dynamic analog of plasma flowing through a spherical chamber of radially varying surface area. In normalized forms, the ion momentum equation is

iTi

dM d dN dM

d d N d d

1θ ηε

ξ ξ ξ ξ= − − − . (21)

Here the minus sign in the gravitational potential term indicates the radially inward direction of the solar self-gravity. The deviation from the conventional neutral gas treatment of the SIP is introduced through the electric space charge-induced force (first term on right-hand side) effect. The normalized expression for conservation of ion flux density is

i

i

dN dM

N d M d

1 1 20.

ξ ξ ξ+ + = (22)

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The normalizations are defined as follows:

e

e

T,

φθ =

sC2

,ψ

η = een

Nn0

,= iin

Nn0

,=

i

s

vM

c,=

J

r,ξ

λ= sJ

J

c,λ

ω= es

i

Tc

m

1 2

,

=

( )J G1 2

4 ,ω πρΘ=i

T

e

T

T.ε =

The notations φ and ψ respectively stand for the dimensional (unnormalized) values of the plasma electrostatic potential and the self-solar gravitational potential as variables

associated with the GES. The dimensional values of the electron and ion population density

variables are respectively denoted by en and in . Likewise, the dimensional ion fluid velocity variable is represented by the symbol iv . The notation η stands for the normalized variable of the self-solar gravitation potential. The notation iN denotes the normalized

value of the ion particle population density variable. Notation M stands for the ion flow

Mach number.

The notations r and ξ stand for the nonnormalized and normalized radial distance respectively from the heliocenter in spherical co-ordinates. The other notations J ,λ sc and

Jω defined as above stand for the Jeans length, sound speed and Jeans frequency respectively. Finally, the notation Tε as defined above stands for the ratio of ion to electron temperature. The ion flux density conservation (eq. 22) contains a term that includes the

effect of geometry on the ion flow dynamics of the SIP mass, self-gravitationally confined in

a spherical region, whose size is to be determined from our own model calculations.

Equations (21) and (22) can be combined to yield a single expression representing the well-

known steady state hydrodynamic flow,

( )T TdM d d

MM d d d

2 1 2 .θ η

ε εξ ξ ξ ξ

− = − + − (23)

There is an obvious difference in the above equation from the corresponding momentum equation under the neutral gas approximation for the SIP. The difference appears, as discussed above, in the form of a space charge effect originating from the Coulomb force on a collective scale (first term on the right-hand side of eq. (21)). The gravito-electrostatic Poisson equations complement the steady dynamical equation (23) for a complete description of the gravito-electrostatic sheath structure, which is formed inside the non-rigid SSB. This is important to emphasize that in the case of a real physical boundary, the plasma sheath is always formed both inside and outside the boundary surface in its close vicinity [12]. The normalized forms of the gravitational and electrostatic Poisson equations for the SWP description are respectively given by

id d

Nd d

2

2

2η η

ξ ξ ξ+ = , and (24)

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Dee i

J

d dN N

d d

22

2

2λ θ θ

λ ξ ξ ξ

+ = − . (25)

Here ( )De eT n e1 22

04λ π= denotes the plasma electron Debye length of the defined SIP system. The other quantities are as defined above as usual. Equations (21)–(25) constitute a

completely closed set of basic governing equations with which to discuss the basic physics

of the GES-potential distribution on the bounded scale. Of course, the discussion also

includes the associated ambipolar radial flow variation of the SIP towards an unknown SSB

which we have to determine self-consistently in this problem with GES-based theory. For a

typical value eT610= K, one can estimate that De J

2010λ λ −≈ which implies that the Debye length is quite a bit smaller than the Jeans scale length of the solar plasma mass. Thus, on

the typical gravitational scale length of the inertially bounded plasma, the limit De J 0λ λ → represents a realistic (physical) approximation. By virtue of this limiting condition, the

entire SIP extending up to the solar boundary and beyond obeys the plasma approximation.

Thus, the quasi-neutrality condition as given below holds good

e iN N N e .θ= = = (26)

This is to note that equation (26) does not mean that the plasma ions are Boltzmannian in thermal character, but inertial species. Equation (26) can be differentiated once in space and further rewritten as,

dN d

N d d

1.

θ

ξ ξ= (27)

By virtue of the plasma approximation, one can justify that the GES of the SIP origin should behave as a quasi-neutral space charge sheath on the Jeans scale size order. The formation mechanism of the defined GES, however, is the same as in the case of plasma-wall interaction process in laboratory confined plasmas. From equations (26)-(27), it is clear that for the electrostatic potential and its gradient being negative, causes the exponential decrease of the plasma density. Finally, the reduced form of the basic set of autonomous closed system of coupled nonlinear dynamical evolution equations under quasi-neutral plasma approximation is enlisted as follows

( )T TdM d d

MM d d d

2 1 2 ,θ η

ε εξ ξ ξ ξ

− = − + − (28)

d dM

d M d

1 20,

θ

ξ ξ ξ+ + = and (29)

d d

ed d

2

2

2.θ

η η

ξ ξ ξ+ = (30)

This set of differential evolution equations constitutes a closed dynamical system of

governing hydrodynamic equations that will be used to determine the existence of a

bounded GES structure on the order of the Jeans scale length ( )Jλ in our GES-model of the

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subsonic origin of the SWP of current interest. Thus the solar parameters ( )M ,ξ ( )sg ξ and ( )θ ξ representing the equilibrium Mach number, solar self-gravity and electrostatic

potential, respectively, will characterize the gravito-electrostatic acoustics in our approach.

5.3 Theoretical analysis of solar surface boundary 5.3.1 Analytical calculations

We first wish to specify the overall condition for the existence of the SSB. Such existence

demands the possibility of a self-consistent bounded solution for the solar self-gravity. The

boundary will correspond to a maximum value of the solar self-gravity at some radial

distance from the heliocenter. This defines a self-consistent location of the SSB. Before we

proceed further, let us argue that the radially outward pulling bulk force effect of the GES-

associated potential term in equation (28) demands a negative electrostatic potential

gradient, that is, d d 0θ ξ < . This makes some physical sense because the ion fluid has to overcome the gravitational barrier to create a global-scale flow of the SIP in a quasi-

hydrostatic way.

Now, if we invoke the concept of exact hydrostatic formation under gravito-electrostatic

force balancing ( )d d d dθ ξ η ξ≈ , the surface potential can be solved to get θ θ η ηΘ Θ− ≈ − . Here the unknown boundary values of θ θΘ= , η ηΘ= and SSBM M= are to be self-consistently specified numerically. The notation ( )SSBM stands for the Mach value associated with the SIP flow at the SSB. Now, by the exact hydrostatic equilibrium condition

in the set of equations (28)-(30), one can get the following set of equations for the SSB

description:

( )T TdM

MM d

2 1 2ε εξ ξ

− = , (31)

d dM

d M d

1 20

η

ξ ξ ξ− + + = , and (32)

d d

ed d

2

2

2 θη η

ξ ξ ξ+ = . (33)

For purpose of the GES analysis, we define the solar self-gravitational acceleration as

sg d dη ξ= . Equation (34) thus reads

s sdg

g ed

2 θξ ξ

+ = . (34)

Finally, the SIP and hence, the SSB are described and specified in terms of the relevant solar

plasma parameters ( )M ,ξ ( )sg ξ and ( )θ ξ representing respectively the equilibrium Mach number, solar self-gravity and electrostatic potential as a coupled dynamical system

of the closed set of equations recast as the following Solar self-gravity equation:

s sdg

g ed

2,θ

ξ ξ+ = (34a)

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Ion continuity equation:

d dM

d M d

1 20,

θ

ξ ξ ξ+ + = and (34b)

Ion momentum equation:

( ) sdM

M gM d

2 1 2 ,α αξ ξ

− = − (34c)

where ( )T i eT T1 1α = +∈ = + , eT is the electron temperature and iT is the inertial ion temperature for the bounded solar plasma on the SIP-scale as already mentioned.

Let us now denote the maximum value ( )gΘ of solar gravity at some radial position ξ ξΘ=where θ θΘ= and apply the necessary condition for the maximization of sg at a spatial coordinate ξ ξΘ= . This condition ( )sdg d 0ξ ξξ Θ= = when used in equation (34) yields

g e2 θξ Θ−Θ Θ= . However, it is not sufficient to justify the occurrence of the maximum value of

sg until and unless the second derivative of sg is shown to have negative value. To derive the sufficient condition for the maximum value of sg at ξ ξΘ= , let us once spatially differentiate equation (34) to yield

s ssd g dg d

g ed d d

2

2 2

2 2.θ

θ

ξ ξ ξ ξ ξ− + = (35)

Now the condition for the maximization of sg at the location ξ ξΘ= can be discussed by considering d d 0θ ξ < in equation (35) under the exact hydrostatic equilibrium approximation ( )d d d d gθ ξ η ξ Θ≈ = near the solar surface to yield the following inequality

sd g

g e g g ed

2

2 2 2

2 20.θ θ

ξ ξξ ξ ξ

Θ Θ

Θ

Θ Θ Θ

Θ Θ=

= − = − = for ~ 1θΘ − (Figs. 8b, 9b, and 10b). Numerically the location of the SSB is found to lie at ~ 3.5ξΘ that matches with g e2

θξ Θ−Θ Θ= for 1.07θΘ = and g 0.6Θ = . It satisfies the analytically derived inequality (36) too. Now the other two equations (32)-(33) can be

simultaneously satisfied in the SSB only for a subsonic solar plasma ion flow speed if

Mach number gradient acquires some appropriate negative minimum near zero

( )( )M dM d 6~ ~ 10ξ − . It is indeed seen numerically that near the maximum solar self-gravity of the SIP mass, the

first and third terms in equation (32) are almost equal and hence the Mach number gradient

term, which is negative in the close vicinity of the SSB, should be smaller than the other two

terms so as to satisfy equations (31) and (32), simultaneously. Actually, the three equations

(31)-(32) and (34) are solved numerically to describe the SSB of the maximum self-

gravitational potential barrier properly where sg associated with the self SIP mass is maximized.

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5.3.2 Numerical calculations

Determination of the autonomous set of the initial values of the defined physical variables is

a prerequisite to solve the nonlinear dynamical evolution equations (34a)-(34c) in general as

an initial value problem. The initial values of the physical variables like ( ) ( )sM g,ξ ξ and ( )θ ξ are defined inside the solar interior and are determined on the basis of extreme

condition of the nonlinear stability analysis [4]. The self-consistent choice of the initial values

is obtained by putting ii

dM d e 2θξ

ξ = − , i

sdg d 0ξξ = and id d 0

ξθ ξ = in these three

equations (34a)-(34c). But the realistic SWP model demands that i

d d 0.ξ

θ ξ ≠ Finally, we determine the expressions for a physically valid set of the initial values of the given physical

variables as follows,

ii iM e21

2θξ= (37)

isi ig e1

2θξ= (38)

This is to note that the initial values of iθ and iξ are chosen arbitrarily. As discussed later, we find that the SSB acquires a negative potential bias ( s ~ 1θ − ) of about -1 kV. It also acquires the maximum value of solar interior gravity ( g ~0.6Θ ) and minimum value of non-zero SIP flow speed ( SSBM

7~ 10− ) at the SSB. The value of the electrostatic potential gradient

at the surface comes out to be ~ –0.6. This means that the strength of the GES-associated

solar surface gravity and electrostatic potential gradient is almost equal. As a result the SSB

is defined by some constant values of the physical variables ( sg M, ,θ ). The SSB values of these parameters are determined through spatial evolution of the coupled system of

equations (34a)-(34c) from the given initial values (37)-(38) inside the SIP zone.

We have used the well-known fourth-order Runge-Kutta method (RK 4) for numerical

solutions of equations (34a)-(34c). By numerical analysis (Figs 8-11), we find that the solar

radius is equal to thrice of the Jeans length ( )Jλ for mean solar mass density ( )g cm 31.41 .ρ −Θ = . From this observation one can easily estimate that j R /3.5λ Θ= . Now comparing our own theoretical value of the solar mass self-gravity with that of the standard

value, we arrive at the following relationship between the solar plasma sound speed ( )sc and the Jeans length ( )Jλ

( )s Jc2 40.6 2.74 10λ = × cm/sec2. (39)

By substituting the value of the Jeans length expressed in terms of the solar radius, one can

determine and specify the mean value of the electron temperature, which is eT ~ 107 K. The sound speed in the SIP under the cold ion model approximation is thus obtained as

sc7~ 3 10× cm s-1. Note that the SWP speed at 1 AU is fixed by the sound speed of the SWP

medium, which is determined and specified by the requirement that a transonic transition

solution occurs on the unbounded scale of the SWP dynamics description.

The gravito-acoustic coupling coefficient could be estimated as g a i em g R T ~ 2.0− Θ ΘΓ = . In absence of the gravitational force, the bulk SIP ion fluid will acquire the flow speed

corresponding to M ~ 1.41 for a negative potential drop of the order of ( )eT e− over a

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distance equal to that of the solar radius. If one estimates the value of gravito-acoustic

coupling coefficient at this velocity defined by M ~ 1.41 , it comes out to be unity. Thus the

quasi-hydrostatic type of equilibrium gravitational surface confinement of the SIP is

ensured. The GES-potential induced outward flow of the SIP is also ensured. Due to

comparable strength of the solar surface gravitational effect of deceleration, the net SIP ion

fluid flow is highly suppressed to some minimum value (~1.0-3.0 cm/sec) corresponding to

M 7~ 10− at the SSB. An interesting point to note here is that near the defined SSB, the electrostatic potential gradient terminates into an almost linear type of profile. The value of its gradient value will provide an estimate of the second order derivative’s contribution into the electrostatic potential which measures the level of local charge imbalance near the solar surface. From our computational plots (Figs. 8b, 9b and 10b), this local charge imbalance comes out to be of the order –0.17, which is equivalent to 17% ion excess charge distributed over a region of size on the order of the plasma Debye sheath scale length. However, the same level of the electrostatic local charge imbalance on the Jeans scale length does not require the inclusion of the role of the Poisson term for the evolution of the electrostatic potential’s profile under the GES-model. Hence, in this sense the GES is practically equivalent to a quasi-neutral plasma sheath with its potential profile tailored and shaped by the potential barrier of the self-gravity of the SIP mass distribution.

5.3.3 Properties of solar surface boundary

Table I lists the defined initial values of the physical variables ( sg , θ , M ) as already discussed and their corresponding boundary values numerically obtained for the

description of the desired SSB. The initial values of sg , θ , and M are associated with the normalized mean SIP mass density, enclosed within a tiny spherical globule having

normalized radius equal to an arbitrarily chosen value of iξ .

Parameter At the Initial

Radial Point ( )iξ At the Solar Surface

Boundary ( )ξΘ Initial Values

Potential θ dd

0θ

ξ=

d

d~ 0.62

θ

ξ− , ~ 1.00θΘ − i

θ , arbitrarily chosen

Gravity sg sdg

d0

ξ= s

dg

d0

ξ= , g ~ 0.60Θ isi ig e

1

2θξ= , derived

Mach

number M i

dMe

d2θ

ξ= −

dM

d0

ξ= , SWPM

7~ 10− ii iM e

21

2θξ= , derived

Table 1. Initial and Boundary Values of Relevant Solar Parameters

From the numerical plots shown in Figs. 8-10, we find that the minimum Mach number

( )SSBM at the specifically defined SSB comes out to be of order 10-7. For this value of Mach number, equation (31) can be simplified to show that near the boundary,

dM d M 83 10 ~ 0ξ ξ −Θ≈ − = − × . This corresponds to a quasi-hydrostatic type of the SSB equilibrium. It arises from gravito-electrostatic balancing with an outward flow of the SIP

having a minimum speed of about 1-3 cm s-1. With these inferences one can argue that the

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SWP originates by virtue of the interaction of the SIP with the SSB. Hence an interconnection

between the Sun and the SWP can be observed by applying the GES model. Here the

boundary is not sharp but distributed over the entire region of the solar interior volume.

The basic principles of the GES coupling govern the solar surface emission process of the

subsonic SWP.

As depicted in table I, the time-independent solar sg -profile associated with the SIP mass

distribution terminates into a diffuse surface boundary. This is characterized and defined by

the quasi-hydrostatic equilibrium sg g d d~ θ ξΘ= , which occurs at about ~ 3.5ξ ξΘ= (see Figs. 8-10). As such, the basic physics of the subsonic origin of the SWP from the SSB is

correlated with the bulk SIP dynamics. We note that the precise definition of the SSB

influences the SWP velocity at 1 AU. Other models report similar observations too [3, 31-41].

The dependence on the ion to electron temperature ratio is quite visible in Fig. 11a. Let us

now discuss the numerical results in the figures individually.

Figure 8 depicts the time-independent profiles of ( sg , θ , M ) and their variations with the ion-to-electron temperature ratio tε for fixed values of the initial point ( i 0.01ξ = ) and plasma sheath potential ( i 0.001θ = − ). As shown in Fig. 8a, the location of the SSB remains the same but its maximum value changes, and a most suitable choice of tε = 0.4 is identified for which the quasi-hydrostatic condition is fulfilled. The E-field profile is invariant for all

chosen values tε =0-0.5. Again, as shown in Fig. 8b, the electrostatic potential corresponding to tε ~ 0.4 comes out to be ~ 1θ θΘ= − (i.e. ~1 kV). Similarly, Fig. 8c depicts the minimum Mach value of SSBM ~10-7 for tε ~0.4 varying by a factor of 2 for other values of tε . Figure 9 depicts the time-independent profiles of sg , θ , and M and their variations with initial position for fixed